Remark 1.3 Note that the results of this section actually hold under the hypotesis that \({\mathcal {F}}\)  admists a global syzygy. This hypothesis is always satisfied, but in the general case the resolution is less obvious. Indeed, following Illusie [1]}, for any sheaf \(\mathcal {F}\)  of \({\mathcal {O}}_X\) -modules on a topological space \(X\) , one can construct the standard free resolution of \(\mathcal {F}\) :
\( \ldots \longrightarrow {\mathcal {R}}({\mathcal {F}})^{2} \stackrel{D^2}{\longrightarrow } {\mathcal {R}}({\mathcal {F}})^{1} \stackrel{D^1}{\longrightarrow } {\mathcal {R}}({\mathcal {F}})^{0} \longrightarrow \mathcal {F} \longrightarrow 0.\) 
